ADVANCES IN MODERN MENTAL CHRONOMETRY EDITED BY : José M. Medina, Willy Wong, José A. Díaz and Hans Colonius PUBLISHED IN : Frontiers in Human Neuroscience 1 June 2015 | Advances in Modern Mental Chronometry Frontiers in Human Neuroscience Frontiers Copyright Statement © Copyright 2007-2015 Frontiers Media SA. All rights reserved. All content included on this site, such as text, graphics, logos, button icons, images, video/audio clips, downloads, data compilations and software, is the property of or is licensed to Frontiers Media SA (“Frontiers”) or its licensees and/or subcontractors. The copyright in the text of individual articles is the property of their respective authors, subject to a license granted to Frontiers. The compilation of articles constituting this e-book, wherever published, as well as the compilation of all other content on this site, is the exclusive property of Frontiers. For the conditions for downloading and copying of e-books from Frontiers’ website, please see the Terms for Website Use. If purchasing Frontiers e-books from other websites or sources, the conditions of the website concerned apply. Images and graphics not forming part of user-contributed materials may not be downloaded or copied without permission. Individual articles may be downloaded and reproduced in accordance with the principles of the CC-BY licence subject to any copyright or other notices. They may not be re-sold as an e-book. As author or other contributor you grant a CC-BY licence to others to reproduce your articles, including any graphics and third-party materials supplied by you, in accordance with the Conditions for Website Use and subject to any copyright notices which you include in connection with your articles and materials. All copyright, and all rights therein, are protected by national and international copyright laws. The above represents a summary only. For the full conditions see the Conditions for Authors and the Conditions for Website Use. ISSN 1664-8714 ISBN 978-2-88919-566-4 DOI 10.3389/978-2-88919-566-4 About Frontiers Frontiers is more than just an open-access publisher of scholarly articles: it is a pioneering approach to the world of academia, radically improving the way scholarly research is managed. The grand vision of Frontiers is a world where all people have an equal opportunity to seek, share and generate knowledge. Frontiers provides immediate and permanent online open access to all its publications, but this alone is not enough to realize our grand goals. Frontiers Journal Series The Frontiers Journal Series is a multi-tier and interdisciplinary set of open-access, online journals, promising a paradigm shift from the current review, selection and dissemination processes in academic publishing. All Frontiers journals are driven by researchers for researchers; therefore, they constitute a service to the scholarly community. At the same time, the Frontiers Journal Series operates on a revolutionary invention, the tiered publishing system, initially addressing specific communities of scholars, and gradually climbing up to broader public understanding, thus serving the interests of the lay society, too. Dedication to Quality Each Frontiers article is a landmark of the highest quality, thanks to genuinely collaborative interactions between authors and review editors, who include some of the world’s best academicians. Research must be certified by peers before entering a stream of knowledge that may eventually reach the public - and shape society; therefore, Frontiers only applies the most rigorous and unbiased reviews. Frontiers revolutionizes research publishing by freely delivering the most outstanding research, evaluated with no bias from both the academic and social point of view. By applying the most advanced information technologies, Frontiers is catapulting scholarly publishing into a new generation. What are Frontiers Research Topics? Frontiers Research Topics are very popular trademarks of the Frontiers Journals Series: they are collections of at least ten articles, all centered on a particular subject. With their unique mix of varied contributions from Original Research to Review Articles, Frontiers Research Topics unify the most influential researchers, the latest key findings and historical advances in a hot research area! Find out more on how to host your own Frontiers Research Topic or contribute to one as an author by contacting the Frontiers Editorial Office: researchtopics@frontiersin.org 2 June 2015 | Advances in Modern Mental Chronometry Frontiers in Human Neuroscience Mental chronometry encompasses all aspects of time processing in the nervous system and constitutes a standard tool in many disciplines including theoretical and experimental psychology and human neuroscience. Mental chronometry has represented a fundamental approach to elucidate the time course of many cognitive phenomena and their underlying neural circuits over more than a century. Nowadays, mental chronometry continues evolving and expanding our knowledge, and our understanding of the temporal organization of the brain in combination with different neuroscience techniques and advanced methods in mathematical analysis. In research on mental chronometry, human reaction/responses times play a central role. Together with reaction times, other topics in mental chronometry include vocal, manual and saccadic latencies, subjective time, psychological time, interval timing, time perception, internal clock, time production, time representation, time discrimination, time illusion, temporal summation, temporal integration, temporal judgment, redundant signals effect, perceptual, decision and motor time, etc. The aim of this research topic is to provide an overview of the state of the art in this field—its relevance, recent findings, current challenges, perspectives and future directions. Thus, as a result, a collection of 14 original research and opinion papers from different experts have been gathered together in a single volume. We hope this research topic will provide a useful framework and an up-to-date set of papers for further discussion on mental chronometry within the human brain. We are grateful to all the referees for their valuable support, effort, and time during the creation of the research topic. Citation: Medina, J. M., Wong, W., Díaz, J. A., Colonius, H., eds. (2015). Advances in Modern Mental Chronometry. Lausanne: Frontiers Media. doi: 10.3389/978-2-88919-566-4 ADVANCES IN MODERN MENTAL CHRONOMETRY Topic Editors: José M. Medina, Universidad de Granada, Spain Willy Wong, University of Toronto, Canada José A. Díaz, Universidad de Granada, Spain Hans Colonius, Carl von Ossietzky Universität Oldenburg, Germany Image courtesy of Dr. José M Medina and Dr. José A Díaz, University of Granada, Spain. Copyright José M Medina and José A Díaz. 3 June 2015 | Advances in Modern Mental Chronometry Frontiers in Human Neuroscience Table of Contents 05 Advances in modern mental chronometry José M. Medina, Willy Wong, José A. Díaz and Hans Colonius Power laws 08 Multifractal analyses of human response time: potential pitfalls in the interpretation of results Espen A. F. Ihlen 12 A theory of power laws in human reaction times: insights from an information- processing approach José M. Medina, José A. Díaz and Kenneth H. Norwich 16 Spectral convergence in tapping and physiological fluctuations: coupling and independence of 1/f noise in the central and autonomic nervous systems Lillian M. Rigoli, Daniel Holman, Michael J. Spivey and Christopher T. Kello 26 What does scalar timing tell us about neural dynamics? Harel Z. Shouval, Marshall G. Hussain Shuler, Animesh Agarwal and Jeffrey P. Gavornik Reaction time distributions 32 Sequential sampling model for multiattribute choice alternatives with random attention time and processing order Adele Diederich and Peter Oswald 45 Manual choice reaction times in the rate-domain Christopher M. Harris, Jonathan Waddington, Valerio Biscione and Sean Manzi 62 A new perspective on binaural integration using response time methodology: super capacity revealed in conditions of binaural masking release Jennifer J. Lentz, Yuan He and James T. Townsend 78 Modeling violations of the race model inequality in bimodal paradigms: co-activation from decision and non-decision components Michael Zehetleitner, Emil Ratko-Dehnert and Hermann J. Müller Human vision system 93 Attentional spreading to task-irrelevant object features: experimental support and a 3-step model of attention for object-based selection and feature-based processing modulation Detlef Wegener, Fingal Orlando Galashan, Maike Kathrin Aurich and Andreas Kurt Kreiter 4 June 2015 | Advances in Modern Mental Chronometry Frontiers in Human Neuroscience 107 Visual evoked potentials to change in coloration of a moving bar Carolina Murd, Kairi Kreegipuu, Nele Kuldkepp, Aire Raidvee, Maria Tamm and Jüri Allik Human auditory system and time perception 115 Overestimation of the second time interval replaces time-shrinking when the difference between two adjacent time intervals increases Yoshitaka Nakajima, Emi Hasuo, Miki Yamashita and Yuki Haraguchi 127 Cortical activity associated with the detection of temporal gaps in tones: a magnetoencephalography study Takako Mitsudo, Naruhito Hironaga and Shuji Mori 138 Temporal dysfunction in traumatic brain injury patients: primary or secondary impairment? Giovanna Mioni, Simon Grondin and Franca Stablum 150 Does time ever fly or slow down? The difficult interpretation of psychophysical data on time perception Miguel A. García-Pérez EDITORIAL published: 06 May 2015 doi: 10.3389/fnhum.2015.00256 Frontiers in Human Neuroscience | www.frontiersin.org May 2015 | Volume 9 | Article 256 | Edited and reviewed by: Hauke R. Heekeren, Freie Universität Berlin, Germany *Correspondence: José M. Medina, jmedinaru@cofis.es Received: 26 February 2015 Accepted: 21 April 2015 Published: 06 May 2015 Citation: Medina JM, Wong W, Díaz JA and Colonius H (2015) Advances in modern mental chronometry. Front. Hum. Neurosci. 9:256. doi: 10.3389/fnhum.2015.00256 Advances in modern mental chronometry José M. Medina 1 *, Willy Wong 2 , José A. Díaz 1 and Hans Colonius 3 1 Departamento de Óptica, Facultad de Ciencias, Universidad de Granada, Granada, Spain, 2 Department of Electrical and Computer Engineering, Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, ON, Canada, 3 Department für Psychologie, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany Keywords: mental chronometry, reaction time, timing and time perception, sensory perception, cognition, human performance, stochastic processes, decision making Mental chronometry encompasses all aspects of time processing in the nervous system and constitutes a standard tool in many disciplines including theoretical and experimental psychology and human neuroscience. Mental chronometry has represented a fundamental approach to elucidate the time course of many cognitive phenomena and their underlying neural circuits over more than a century. Nowadays, mental chronometry continues evolving and expanding our knowledge, and our understanding of the temporal organization of the brain in combination with different neuroscience techniques and advanced methods in mathematical analysis. In research on mental chronometry, human reaction/responses times (RT) play a central role. Together with RTs, other topics in mental chronometry include vocal, manual and saccadic latencies, subjective time, psychological time, interval timing, time perception, internal clock, time production, time representation, time discrimination, time illusion, temporal summation, temporal integration, temporal judgment, redundant signals effect, perceptual, decision and motor time, etc. It is worth noting that there have been well over 37,000 full-length journal papers published in the last decade on a variety of topics related to simple and choice RTs, etc. This amounts to approximately 3800 papers per year, or roughly 10 papers per day (source: PubMed, similarly Thomson Reuters Web of Science). There are comprehensive reviews that deal extensively with the history of mental chronometry, experimental methods and paradigms, stochastic models, etc. as well as its relationship to other psychological and physiological variables, neuroscience methods and clinical applications (Laming, 1968; Posner, 1978, 2005; Welford and Brebner, 1980; Townsend and Ashby, 1983; Luce, 1986; Meyer et al., 1988; Robbins and Brown, 1990; Schall, 2001; Mauk and Buonomano, 2004; Smith and Ratcliff, 2004; Jensen, 2006; Gold and Shadlen, 2007; Linden, 2007; Grondin, 2010; Merchant et al., 2013; Allman et al., 2014). The aim of this research topic is to provide an overview of the state of the art in this field—its relevance, recent findings, current challenges, perspectives and future directions. Thus, as a result, a collection of 14 original research and opinion papers from different experts have been gathered together in a single volume. They outline a selection of unsolved problems and topics in mental chronometry mainly within the context of the human visual system as well as the auditory system. One of the unsolved problems is the functional role of power laws in RT variability and in the study of timing. Power laws are ubiquitous in many complex systems, and their experimental validity and theoretical support represent a fundamental aspect in many disciplines, such as in biology, physics, finance, etc. In this theme issue, the papers of Ihlen (2014), Medina et al. (2014), Rigoli et al. (2014) and Shouval et al. (2014) address different aspects of power laws, namely, multifractal analysis on RT series; an information theoretic basis of RT power law scaling; Fourier-based power law correlations (“1/ f noise”) in a tapping task and its comparison with other physiological processes (e.g., heartbeat intervals); and a log-power law model of the firing rate of neurons in interval timing. A second unsolved problem involves RT-based methods and research into RT distributions. RT distributions are typically positively skewed and often exhibit long right-tails in the time-domain. 5 Medina et al. Advances in modern mental chronometry A long-standing issue deals with the shape of RT distributions, their intrinsic stochastic latency mechanisms and neural basis. Sequential-sampling models are a common approach widely used in human RTs and simple decision making (Smith and Ratcliff, 2004). Diederich and Oswald present a RT sequential- sampling model for multiple stimulus features based on an Ornstein–Uhlenbeck diffusion process (Diederich and Oswald, 2014). In a different type of analysis, the work of Harris et al. introduces an alternative approach to examine very long RTs in the rate-domain (i.e., 1/RT). These authors investigate the shape of choice RT distributions and sequential correlations using autoregressive techniques (Harris et al., 2014). In general, RT distributions exhibit faster RTs under summation/facilitation tasks when two or more redundant signals are available as compared with a single signal or sensory modality (e.g., binocular vs. monocular vision), usually called redundant signals effect. The work of Lentz et al. examines binaural vs. monaural hearing performance under noise masking tasks using modeling techniques based on the concept of workload capacity and different processing mechanisms (e.g., serial vs. parallel, etc.) and stopping rules (Lentz et al., 2014). Within the same redundant signals paradigm, Zehetleitner et al. study bimodal (audio-visual) facilitation effects using sequential-sampling models (Zehetleitner et al., 2015). Regarding the human vision system, the work of Wegener et al. examines the visual attention mechanisms using colored stimuli (random dot patterns), and they have presented a novel three-step model of attention to predict the corresponding RT distributions (Wegener et al., 2014). The work of Murd et al. exemplifies the used RTs in conjunction with visual evoked potentials in the detection of visual colored stimuli (Murd et al., 2014). There are also studies focusing on the auditory system, including the work of Nakajima et al. that investigates the foundations of time perception using a time illusion based on an overestimation of a second time interval preceded by a first time interval or time-shrinking effect (Nakajima et al., 2014). Mitsudo et al. present recorded magnetoencephalogram signals in tasks that require to judge temporal gaps in tones and have discussed their implications in the organization of the auditory cortex (Mitsudo et al., 2014). Within the same time perception paradigm, Mioni et al. show a detailed review on temporal dysfunctions in traumatic brain injury patients (Mioni et al., 2014). The present theme issue also includes the work of García-Pérez who introduces a unified model to analyze different psychophysical tasks in time perception and estimation of the psychometric function (García-Pérez, 2014). We hope this research topic will provide a useful framework and an up-to-date set of papers for further discussion on mental chronometry within the human brain. Acknowledgments We acknowledge all the referees for their valuable support, effort, and time. References Allman, M. J., Teki, S., Griffiths, T. D., and Meck, W. H. (2014). Properties of the internal clock: first- and second-order principles of subjective time. Annu. Rev. Psychol. 65, 743–777. doi: 10.1146/annurev-psych-010213- 115117 Diederich, A., and Oswald, P. (2014). Sequential sampling model for multiattribute choice alternatives with random attention time and processing order. Front. Hum. Neurosci. 8:697. doi: 10.3389/fnhum.2014.00697 García-Pérez, M. A. (2014). Does time ever fly or slow down? The difficult interpretation of psychophysical data on time perception. Front. Hum. Neurosci . 8:415. doi: 10.3389/fnhum.2014.00415 Gold, J. I., and Shadlen, M. N. (2007). The neural basis of decision making. Annu. Rev. Neurosci. 30, 535–574. doi: 10.1146/annurev.neuro.29.051605. 113038 Grondin, S. (2010). Timing and time perception: a review of recent behavioral and neuroscience findings and theoretical directions. Atten. Percept. Psychophys. 72, 561–582. doi: 10.3758/APP.72.3.561 Harris, C., Waddington, J., Biscione, V., and Manzi, S. (2014). Manual choice reaction times in the rate-domain. Front. Hum. Neurosci. 8:418. doi: 10.3389/fnhum.2014.00418 Ihlen, E. A. F. (2014). Multifractal analyses of human response time: potential pitfalls in the interpretation of results. Front. Hum. Neurosci. 8:523. doi: 10.3389/fnhum.2014.00523 Jensen, A. R. (2006). Clocking the Mind: Mental Chronometer Individual Differences . Amsterdam: Elsevier. Laming, D. (1968). Information Theory of Choice-reaction Times. San Diego, CA: Academic Press. Lentz, J., He, Y., and Townsend, J. T. (2014). A new perspective on binaural integration using response time methodology: super capacity revealed in conditions of binaural masking release. Front. Hum. Neurosci. 8:641. doi: 10.3389/fnhum.2014.00641 Linden, D. E. (2007). What, when, where in the brain? Exploring mental chronometry with brain imaging and electrophysiology. Rev. Neurosci. 18, 159–171. doi: 10.1515/REVNEURO.2007.18.2.159 Luce, R. D. (1986). Response Times. New York, NY: Oxford University Press. Mauk, M. D., and Buonomano, D. V. (2004). The neural basis of temporal processing. Annu. Rev. Neurosci. 27, 307–340. doi: 10.1146/annurev.neuro.27.070203.144247 Medina, J. M., Díaz, J. A., and Norwich, K. (2014). A theory of power laws in human reaction times: insights from an information-processing approach. Front. Hum. Neurosci. 8:621. doi: 10.3389/fnhum.2014.00621 Merchant, H., Harrington, D. L., and Meck, W. H. (2013). Neural basis of the perception and estimation of time. Annu. Rev. Neurosci. 36, 313–336. doi: 10.1146/annurev-neuro-062012-170349 Meyer, D. E., Osman, A. M., Irwin, D. E., and Yantis, S. (1988). Modern mental chronometry. Biol. Psychol. 26, 3–67. doi: 10.1016/0301-0511(88)90013-0 Mioni, G., Grondin, S., and Stablum, F. (2014). Temporal dysfunction in traumatic brain injury patients: primary or secondary impairment? Front. Hum. Neurosci. 8:269. doi: 10.3389/fnhum.2014.00269 Mitsudo, T., Hironaga, N., and Mori, S. (2014). Cortical activity associated with the detection of temporal gaps in tones: a magnetoencephalography study. Front. Hum. Neurosci. 8:763. doi: 10.3389/fnhum.2014.00763 Murd, C., Kreegipuu, K., Kuldkepp, N., Raidvee, A., Tamm, M., and Allik, J. (2014). Visual evoked potentials to colour change of a moving bar. Front. Hum. Neurosci. 8:19. doi: 10.3389/fnhum.2014.00019 Nakajima, Y., Hasuo, E., Yamashita, M., and Haraguchi, Y. (2014). Overestimation of the second time interval replaces time-shrinking when the difference between two adjacent time intervals increases. Front. Hum. Neurosci. 8:281. doi: 10.3389/fnhum.2014.00281 Posner, M. I. (1978). Chronometric Explorations of Mind. Oxford: Lawrence Erlbaum. Posner, M. I. (2005). Timing the brain: mental chronometry as a tool in neuroscience. PLoS Biol. 3:e51. doi: 10.1371/journal.pbio.0030051 Frontiers in Human Neuroscience | www.frontiersin.org May 2015 | Volume 9 | Article 256 | 6 Medina et al. Advances in modern mental chronometry Rigoli, L. M., Holman, D., Spivey, M., and Kello, C. (2014). Spectral convergence in tapping and physiological fluctuations: coupling and independence of 1/f noise in the central and autonomic nervous systems. Front. Hum. Neurosci. 8:713. doi: 10.3389/fnhum.2014.00713 Robbins, T. W., and Brown, V. J. (1990). The role of the striatum in the mental chronometry of action: a theoretical review. Rev. Neurosci. 2, 181–214. doi: 10.1515/REVNEURO.1990.2.4.181 Schall, J. D. (2001). Neural basis of deciding, choosing and acting. Nat. Rev. Neurosci. 2, 33–42. doi: 10.1038/35049054 Shouval, H. Z., Hussain Shuler, M. G., Agarwal, A., and Gavornik, J. P. (2014). What does scalar timing tell us about neural dynamics? Front. Hum. Neurosci. 8:438. doi: 10.3389/fnhum.2014.00438 Smith, P. L., and Ratcliff, R. (2004). Psychology and neurobiology of simple decisions. Trends Neurosci. 27, 161–168. doi: 10.1016/j.tins.2004. 01.006 Townsend, J. T., and Ashby, F. G. (1983). The Stochastic Modeling of Elementary Psychological Processes. Cambridge: Cambridge University Press. Wegener, D., Galashan, F. O., Aurich, M. K., and Kreiter, A. K. (2014). Attentional spreading to task-irrelevant object features: experimental support and a 3-step model of attention for object-based selection and feature-based processing modulation. Front. Hum. Neurosci. 8:414. doi: 10.3389/fnhum.2014.00414 Welford, A. T., and Brebner, J. M. T. (1980). Reaction Times. New York, NY: Academic Press. Zehetleitner, M., Ratko-Dehnert, E., and Mueller, H. J. (2015). Modeling violations of the race model inequality in bimodal paradigms: co-activation from decision and non-decision components. Front. Hum. Neurosci. 9:119. doi: 10.3389/fnhum.2015.00119 Conflict of Interest Statement: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Copyright © 2015 Medina, Wong, Díaz and Colonius. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Frontiers in Human Neuroscience | www.frontiersin.org May 2015 | Volume 9 | Article 256 | 7 OPINION ARTICLE published: 21 July 2014 doi: 10.3389/fnhum.2014.00523 Multifractal analyses of human response time: potential pitfalls in the interpretation of results Espen A. F. Ihlen* Department of Neuroscience, Norwegian University of Science and Technology, Trondheim, Norway *Correspondence: espen.ihlen@ntnu.no Edited by: José M. Medina, Universidad de Granada, Spain Reviewed by: Fred Hasselman, Radboud University Nijmegen, Netherlands Helmut Ahammer, Medical University of Graz, Austria Keywords: response times, 1/ f noise, multifractal, variability, long-range dependency, fractal INTRODUCTION Analyses of response time series have provided insight into mental organiza- tion and cognitive processes used in a wide variety of tasks such as simple reac- tion time, word naming, choice decision, visual search, memory search, and lexi- cal decision (Gilden, 2001). One of the new and frequently used sets of analy- ses is the numerical definition of scale invariant structure of response time series, also called 1/ f fluctuations. Component- oriented theories suggest that this scale invariant structure originated from an idiosyncratic mechanism in the cognitive system, whereas interaction-oriented the- ories argue that scale invariant structure in response time series arises from self- organizing interaction between different sources and mechanisms (cf. Diniz et al., 2011). In this short commentary, new analyses of human response time called multifractal analyses will be introduced, and potential pitfalls of interpreting the results of these analyses will be discussed. Multifractal analyses quantify the inter- mittent structure of response time series that are created by interactions between temporal scales of response series (Ihlen and Vereijken, 2010, 2013). Even though these analyses have been recently intro- duced in analysis of human behavior, their mathematical fundament of these analyses was introduced four decades ago (Yaglom, 1966; Mandelbrot, 1974). Typically, response time series with a large number of trials will contain intermit- tent periods with a higher number of slow response latencies than the rest of the response series (e.g., Holden et al., 2009). These intermittent periods of slow response latencies might indicate shifts in the participant attention to the stim- uli source or active periods of response error corrections (Ihlen and Vereijken, 2010, 2013). In order to quantify the intermittent structure of response time series, multifractal analyses combine two fundamental classes of analyses: (1) model based analyses of the response time distri- bution and (2) analyses of the dependency of the time ordering of the responses. Class 1 analyses have shown that the response time distributions across cogni- tive tasks is unimodal, positively skewed, and with a heavy right tail containing the slow response latencies (e.g., Luce, 1986; Holden et al., 2009). Class 2 anal- yses have shown that the response times have long-range dependency across hun- dreds and even thousands of trials and, consequently, that the response time series cannot be considered to be independent random variables assumed by class 1 analyses (Gilden, 2001). The long-range dependency (i.e., monofractal structure) of the response time series are numeri- cal, defined as a single scaling exponent by spectral analyses, autocorrelation anal- yses, detrended fluctuation analysis, and dispersion analysis, to mention but a few (cf. Diniz et al., 2011). However, Class 2 analyses assume that the response time is Gaussian distributed, whereas Class 1 analyses indicate that they have a non- Gaussian heavy tail toward slow response latencies. Multifractal analyses are able to parameterize the non-Gaussian heavy tails that are created using intermittent varia- tion by assessing the complete spectrum of scaling exponents. Thus, multifrac- tal analyses are important extensions of monofractal analyses of response time series. All multifractal analyses are based on a decomposition of the response time series into a scale-dependent measure that iden- tifies the periods of intermittent varia- tion (see upper panel of Figure 1 ). The scale dependent measure is the basis for computation of the multifractal spectra along two formalisms (see arrows A and B in Figure 1 ). In the Legendre formal- ism, the scale-dependent measure μ s , t is used in the computation of the q -order moment. μ s , t is amplified by the positive q -orders in the periods with large vari- ation, whereas μ s , t is amplified by the negative q -orders in periods with small variation. An exponent ζ q is then esti- mated from the scaling of each of the q -order moments before the multifractal spectra are computed from ζ q (see Ihlen and Vereijken, 2013 for further details). In the large deviation formalism, local exponents are computed from the scale- dependent measure μ s , t , and the mul- tifractal spectrum is estimated from the distribution of the local exponents. The increased width of multifractal spectra will reflect more distinct periods of intermit- tent variation in response time series (see example in Figure 2 in Ihlen and Vereijken, 2013). Additional surrogate tests also detect the periods influenced by mul- tiplicative interactions between temporal scales (Ihlen and Vereijken, 2010). The different multifractal analyses like struc- ture function approach , entropy analyses , wavelet transformation modulus maxima , Frontiers in Human Neuroscience www.frontiersin.org July 2014 | Volume 8 | Article 523 | HUMAN NEUROSCIENCE 8 Ihlen Potential pitfalls of multifractal analyses FIGURE 1 | A flow chart of the estimation of the multifractal spectrum D h by analyses within the Legendre formalism ( arrows A) and large deviation formalism ( arrows B). The basis for all multifractal analyses within both formalisms is the scale-dependent measure ( upper contour plot ) that decomposes the intermittent variation of response time series into both the time and scale domain. The red contours indicate large scale-dependent measures of the response time series that coincide with the time periods of intermittent large variations. In contrast, the blue contours indicate small scale-dependent measures that coincide with the time periods of intermittent small variations. The panel below the top arrow A indicates that the scale-dependent measure is summarized by its q -order statistical moment. The statistical moments with positive q ’s amplify the large μ s , t (i.e., red contours ) whereas the statistical moments with negative q ’s amplify the small μ s , t (i.e., blue contours ). The scaling exponent ζ q numerically defines the power law relation of the intermittent periods with large (i.e., positive q ’s) and small variation (i.e., negative q ’s). The panel below the bottom arrow A illustrates a multifractal spectrum D h estimated from ζ q . The panel below the top arrow B illustrates the direct estimation of the local singularity exponent h t as the local slope of log( μ s , t ) vs. log( s ) for each time instant t . The panel below the bottom arrow B illustrates the multifractal spectrum D h estimated from the distribution of local singularity exponent h t . Adapted from Ihlen and Vereijken (2013). gradient modulus wavelet projection , and multifractal detrended fluctuation analysis are defined by the particular way the scale- dependent measures are computed (Ihlen, 2013a; Ihlen and Vereijken, 2013). The Legendre and large deviation formalisms contain statistical assessments of multi- fractality. Various geometrical assessments have been suggested in the literature that estimates the box counting dimension of the time series (e.g., Russel et al., 1980; Chaudhuri and Sarkar, 1995). However, these methods are only numerically sta- ble for positive q orders and, consequently, only estimate the left tail of the mul- tifractal spectrum. Technical details for the computation of different multifrac- tal analyses within the Legendre and large deviation formalisms, their parameter set- tings, Matlab codes, and comparison of their performance can be found elsewhere (Kantelhardt et al., 2002; Turiel et al., 2006; Kantelhardt, 2011; Ihlen, 2013a). Multifractal analyses have been applied to several cognitive tasks like simple reac- tion time, word naming, choice decision, and feedback manipulation (Ihlen and Frontiers in Human Neuroscience www.frontiersin.org July 2014 | Volume 8 | Article 523 | 9 Ihlen Potential pitfalls of multifractal analyses Vereijken, 2010; Kuznetsov and Wallot, 2011). All results from these studies indi- cate that response time series have mul- tifractal properties that are not described by conventional monofractal analyses and that some of these properties might be task dependent. POTENTIAL PITFALLS IN THE INTERPRETATION OF MULTIFRACTAL ANALYSES The interpretation of multifractal spectra of response time series has potential pit- falls. First, the multifractal spectra alone do not indicate that intermittent response time variation is generated by interac- tion between temporal scales. Wide mul- tifractal spectra of response time series can reflect a power-law response time dis- tribution and not intermittency gener- ated by multiplicative interactions (Ihlen, 2013b). Surrogate tests have to be used to properly identify multiplicative inter- actions between temporal scales. In these tests, surrogate versions of the response time series are created that eliminate the interaction between temporal scales but preserve all other statistical properties. Multiplicative interaction is present when there is a significant difference between response time series and its surrogate series (e.g., Ihlen and Vereijken, 2010). Second, response time series of 1000 trials might be too small to establish the presence of multifractality. An ideal monofractal signal will have an infinite number of scales whereas the 1000 trials of response series will only give three scales of order (i.e., 10, 100, and 1000 trials). However, in contrast to ideal monofrac- tal signal, a multifractal signal has scale invariant properties only up to a max- imum scale (Bacry et al., 2001). The q -order moments and scale-dependent measure converge into a single point on this maximum scale. Thus, in contrast to monofractal analyses, it is sufficient for multifractal analyses to include scales up to the maximum order. Assuming that the signal originates from a prototypical multifractal process, called a multiplica- tive cascade, the maximum scale could be assessed by analysis of the autocor- relation function (Bacry et al., 2001). Nevertheless, the estimation error of the multifractal spectra related to the num- ber of trials in the response will also be dependent on the chosen q -range for the methods within the Legendre formalism and the unknown degree of multifractality. Large degree of multifractality will need large number of trials for a robust assess- ment of the tails of the multifractal spec- tra. Consequently, multifractal analysis is quite sensitive to differentiate between monofractal and multifractal response time series, but not between response time series with large degree of multifrac- tality. Furthermore, multifractal analysis of moderately sized response time series will both be more susceptible to noise and non-stationarities compared to longer time series (Ihlen, 2013a). A possible solu- tion is to compare the results of two or more multifractal analysis before inter- preting the results. Large deviations in the results of two multifractal analyses indi- cate that response time series deviate from multifractality and that the results from these analyses must be interpreted with caution. Third, no single multifractal analy- sis seems to have superior performance assessing the multifractal spectra of response time series. Previous studies statistical methods based on wavelet transformation, like wavelet transform modulus maxima, has been shown to superior to conventional methods based on the structure function (Muzy et al., 1993). Furthermore, both multifractal detrended fluctuation analysis and gra- dient modulus wavelet projection has shown superior performance to wavelet transform modulus maxima on moder- ate sized time series (Kantelhardt et al., 2002; O ́ swi ̨ ecimka et al., 2006; Turiel et al., 2006). Kelty-Stephen et al. (2013) have suggested that an entropy based analysis is the best method to assess the mul- tifractal spectrum from response time series and that other multifractal analy- ses have inferior performance compared to this method using their choice of a scale-dependent measure. However, recent systematic comparison of multifractal analyses shows that all multifractal analy- ses have different pros and cons and that no single analyses seem to be superior to others (Ihlen, 2013a). Fourth, the origin of multifractal and intermittent variation in response time series is still debated. Intermittent varia- tion in response time has been suggested to be caused by changes in the participants’ attention to stimuli or intermittent error corrections (Ihlen and Vereijken, 2010) and linked to cognitive phenomena like strong anticipation (Stephen and Dixon, 2011). Furthermore, multifractal spectra have been suggested to reflect to a greater extent the presence of self-organization and interaction-dominant dynamics com- pared to the outcomes of conventional monofractal analyses (Ihlen and Vereijken, 2010; Kelty-Stephen et al., 2013). The interaction-dominant view has been sug- gested to contrast explicit models of an idiosyncratic mechanism in the cogni- tive system specific to cognitive tasks or the dynamics of particular localized com- ponents (e.g., Van Orden et al., 2003). However, idiosyncratic mechanisms for multifractal variations have been sug- gested for human locomotion and cardiac function, which indicates that intermit- tent variations can be generated by task specific components (Ivanov et al., 1998; West and Scafetta, 2003). It is unlikely that any analysis or model will provide conclu- sive evidence on the generating processes of multifractal variation in response time series (Hasselman, 2013; cf. Kantz and Schreiber, 2004). The generating processes of multifractal and intermittent varia- tion should be decided by experimenta- tion under conditions of strong inference (Hasselman, 2013). Consequently, exper- imental design should be use to confirm predicted changes in the multifractal spec- tra. Predicted covariation between local scaling exponents of the response time series and other psychological measures will indicate a common generating process of the multifractality of these signals. As an example, intermittent changes in attention and error correction could be verified by multifractal analyses of gaze fixation and eye movements during the same cogni- tive task (e.g., Kelty-Stephen and Mirman, 2013). In summary, caution should be made when inferring response time series as multifractal in a strict mathematical sense. Nevertheless, the width of the multi- fractal spectra could still be a sensitive index of the intermittency of the response time series even though the intermit- tency is not prototypical multifractal. The main advantage of multifractal analyses of response time series is their ability to Frontiers in Human Neuroscience www.frontiersin.org July 2014 | Volume 8 | Article 523 | 10 Ihlen Potential pitfalls of multifractal analyses assess the temporal changes in their scale invariant structure. Further studies should focus on the assessment of generating pro- cesses of multifractal by experimentation under strong inference. This might include the assessment of temporal changes in the local scaling exponent (i.e., the local structure of response time variation) in more heterogeneous and real-life experi- ments where the task conditions and char- acteristics of the stimuli involve change across trials. Furthermore, the correla- tion between the temporal changes in the structure of the response time variation and other neurophysiological and psy- chological measurements can be assess